### Małgorzata Filipczak, Tomasz Filipczak

## On the comparison of the density type topologies generated by sequences and by functions

**Abstract. In the paper we investigate density type topologies generated by functions** *f* satisfying condition lim inf

*x→0*^{+}
*f(x)*

*x*

*>* 0, which are not generated by any sequence.

*2000 Mathematics Subject Classification: 54A10, 26A15, 28A05.*

*Key words and phrases: density points, density topology, comparison of topologies.*

### Through the paper we shall use standard notation: ℝ will be the set of real numbers, *ℕ the set of positive integers, L the family of Lebesgue measurable subsets* of *ℝ and |E| the Lebesgue measure of a measurable set E. By Φ* ^{d} *(E) we shall* *denote the set of all Lebesgue density points of measurable set E (i.e. Φ* *d* *(E) =* n *x* *∈ ℝ; lim* *h* *→0*

^{d}

^{+}

*|(x−h;x+h)∩E|*

*2h* = 1 o

*) and by T* ^{d} the density topology consists of *measurable sets satisfying E ⊂ Φ* ^{d} *(E). For any operators Φ, Ψ : L → L we write* *Φ ⊂ Ψ if Φ (E) ⊂ Ψ (E) for every E ∈ L. If Φ ⊂ Ψ and Φ 6= Ψ then we write Φ ⊈ Ψ.*

^{d}

^{d}

### We will consider two generalizations of Lebesgue density. First of them, called a density generated by a sequence, was introduced by J. Hejduk and M. Filipczak in [6]. For a convenience we will formulate definitions using decreasing sequences tending to zero, instead of nondecreasing sequences going to infinity.

### Let e *S be the family of all decreasing sequences tending to zero. We will denote* sequences from e *S by (a* ^{n} *) or by hai. Let hai ∈ e* *S, E ∈ L and x ∈ ℝ. We shall say* *that x is an hai-density point of E (a right-hand hai-density point of E) if*

^{n}

*n* lim *→∞*

*|E ∩ [x − a* ^{n} *; x + a* *n* *]|*

^{n}

*2a* *n*

### = 1 ( lim

*n* *→∞*

*|E ∩ [x; x + a* ^{n} *]|*

^{n}

*a* *n*

### = 1).

### By Φ _{hai} *(E) (Φ* ^{+} _{hai} *(E)) we will denote the set of all hai-density (right-hand hai-*

_{hai}

_{hai}

*density) points of E. In the same way one may define* *left-hand hai-density point of*

*E and the set Φ* ^{−} _{hai} *(E). Evidently, Φ* _{hai} *(E) = Φ* ^{+} _{hai} *(E) ∩ Φ* ^{−} _{hai} *(E).*

^{−}

_{hai}

_{hai}

_{hai}

^{−}

_{hai}

### In [6] it was proved that Φ _{hai} is a lower density operator and the family *T* *hai* =

_{hai}

*E* *∈ L; E ⊂ Φ* *hai* *(E)*

*is a topology containing the density topology T* ^{d} *. Moreover for a* *n* = _{n} ^{1} , Φ _{hai} = Φ *d*

^{d}

_{n}

_{hai}

*and T* *hai* *= T* ^{d} .

^{d}

*For any pair of sequences hai and hbi from e* *S we denote by hai∪hbi the decreasing* *sequence consisting of all elements from hai and hbi. It is clear that hai ∪ hbi ∈ e* *S* and that

### Proposition 1 *If Φ* _{hai} *⊂ Φ* *hbi* *then Φ* _{hai∪hbi} = Φ _{hai} *.*

_{hai}

_{hai∪hbi}

_{hai}

### The second type of density we will observe are densities generated by functions.

*We denote by A the family of all functions f : (0; ∞) → (0; ∞) such that* (A1) lim

*x→0*

^{+}

*f (x) = 0,* (A2) lim inf

*x* *→0*

^{+}

*f(x)*

*x* *<* *∞,* *(A3) f is nondecreasing.*

*Let f ∈ A. We say that x is a right-hand f-density point of a measurable set E* if

*h→0* lim

^{+}

*|(x; x + h) \ E|*

*f (h)* *= 0.*

### By Φ ^{+} _{f} *(E) we denote the set of all right-hand f-density points of E. In the same* way one may define *left-hand f-density points of E and the set Φ* ^{−} _{f} *(E). We say that* *x is an f* *-density point of E if it is a right and a left-hand f-density point of E. By* Φ *f* *(E) we denote the set of all f-density points of E, i.e. Φ* *f* *(E) = Φ* ^{+} _{f} *(E) ∩ Φ* ^{−} *f* *(E).*

_{f}

^{−}

_{f}

_{f}

^{−}

*For any f ∈ A, the family*

*T* ^{f} *= {E ∈ L; E ⊂ Φ* ^{f} *(E)}*

^{f}

^{f}

### forms a topology stronger than the natural topology on the real line (see [1, Th. 7]

### and [4, Th. 1]).

### Proposition 2 ([4, Prop. 4]) *For each f, g ∈ A, an inclusion T* ^{f} *⊂ T* ^{g} *holds if* *and only if Φ* *f* *⊂ Φ* ^{g} *.*

^{f}

^{g}

^{g}

*In [2] and [3] it has been shown that properties of f-density operator Φ* *f* and *f -density topology* *T* ^{f} are strictly depended on lim inf

^{f}

*x* *→0*

^{+}

*f(x)*

*x* . In our paper we are *interested in topologies generated by functions f ∈ A for which lim inf*

*x→0*

^{+}

*f(x)*

*x* *> 0.*

*The family of all such functions we will denote by A* ^{1} .

*Topologies generated by functions from A* ^{1} *and from A \ A* ^{1} have quite different

### properties (for example they satisfy different separation axioms - see [3, Th. 5 and

*Th. 7)]). On the other hand for any function f from A* ^{1} *properties of T* ^{f} are similar to properties of topologies generated by sequences, so similar to properties of the *density topology T* ^{d} (compare [4, Th. 3], [9], [7] and [8]). Moreover

^{f}

^{d}

### Proposition 3 ([4, Th. 5]) *For any sequence hsi ∈ e* *S, the function f defined by* *a formula*

*f (x) = s* *n* *for x ∈ (s* ^{n+1} *; s* *n* ] *belongs to A* ^{1} *and Φ* _{hsi} = Φ *f* *.*

^{n+1}

_{hsi}

### Corollary 4 *For each hai , hbi ∈ e* *S, an inclusion T* *hai* *⊂ T* *hbi* *holds if and only if* Φ _{hai} *⊂ Φ* *hbi* *.*

_{hai}

*In [4] we have constructed a function f ∈ A* ^{1} such that Φ *f* *6= Φ* *hsi* for each *hsi ∈ e* *S. We will remain the definition of that function, omitting the proof, because* we will construct a similar one in Theorem 11.

### Example 5 ([4, Th. 6]) Let us define sequences *hwi = (2, 2, 3, 3, 3, 4, 4, 4, 4, . . .) ,*

*hri = (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) ,* *a* 0 *= 1, a* *n* = *a* *n* *−1*

*w* ^{2} _{n} *for n 1,* *b* *n* *= a* *n* *w* *n* = *a* *n* *−1*

_{n}

*w* *n* *for n 1.*

### Evidently

*n* lim *→∞*

*a* *n* *−1*

*b* *n*

### = lim

*n* *→∞*

*b* *n*

*a* *n* *= ∞.*

*The function f defined by a formula* *f (x) =*

### *a* *n* *−1* *for x ∈ (b* ^{n} *; a* *n* *−1* ] , *b* *n* *r* *n* *for x ∈ (a* ^{n} *; b* *n* ] . *belongs to A* ^{1} and Φ *f* *6= Φ* *hsi* *for each hsi ∈ e* *S.*

^{n}

^{n}

### The function constructed above proves that the family

*T* ^{f} *; f ∈ A* ^{1} is big- ger than n

^{f}

*T* *hsi* *; hsi ∈ e* *S* o

### . We will show that there exist continuum topologies from

### *T* ^{f} *; f ∈ A* ^{1}

^{f}

*\* n

*T* *hsi* *; hsi ∈ e* *S* o

*and that for any pair of sequences hai , hbi ∈ e* *S satis-* *fying T* *hai* *⊈ T* *hbi* *there is a function f ∈ A* ^{1} *such that T* *hai* *⊈ T* ^{f} *⊈ T* *hbi* *and T* ^{f} *6= T* *hsi*

^{f}

^{f}

*for hsi ∈ e* *S.*

### Set

*f* *α* *(x) = f* * x* *α*

### *and s* *α* *(n) = αs (n)*

*for f ∈ A* ^{1} *, hsi ∈ e* *S and α > 0. Obviously f* ^{α} *∈ A* ^{1} *and hs* ^{α} *i ∈ e* *S.*

^{α}

^{α}

### Proposition 6 *If Φ* *f* = Φ _{hsi} *then Φ* *f*

_{hsi}

*α*

### = Φ _{hs}

_{hs}

_{α}_{i} *.*

_{i}

*Proof It is sufficient to prove that for any measurable set E* *0 ∈ Φ* ^{+} *f*

*α*

*(E) ⇐⇒ 0 ∈ Φ* ^{+} _{hs}

_{hs}

*α*

*i* *(E) .* *If 0 ∈ Φ* ^{+} *f*

*α*

*(E) then*

### *(0; x) ∩* _{α} ^{1} *E* ^{0}

_{α}

^{0}

*f (x)* =

### 1

*α* *|(0; αx) ∩ E* ^{0} *|* *f* *α* *(αx)*

^{0}

*x* *→0+*

*−→ 0,*

*and so 0 ∈ Φ* ^{+} *f* 1

*α* *E* = Φ ^{+} _{hsi} _{α} ^{1} *E* . Hence

_{hsi}

_{α}

*|(0; s* ^{α} *(n)) ∩ E* ^{0} *|* *s* *α* *(n)* =

^{α}

^{0}

### *(0; s (n)) ∩* _{α} ^{1} *E* ^{0} *s (n)*

_{α}

^{0}

*n* *→∞*

*−→ 0,*

*which gives 0 ∈ Φ* ^{+} _{hs}

_{hs}

_{α}_{i} *(E).*

_{i}

*Conversely, if 0 ∈ Φ* ^{+} _{hs}

_{hs}

*α*

*i* *(E) then* *(0; s (n)) ∩* *α* ^{1} *E* ^{0}

^{0}

*s (n)* *= |* *(0; s* *α* *(n)) ∩ E* ^{0} *|* *s* *α* *(n)*

^{0}

*n→∞* *−→ 0,*

*and consequently 0 ∈ Φ* ^{+} _{hsi} *α* ^{1} *E* = Φ ^{+} _{f} _{α} ^{1} *E* . Thus

_{hsi}

_{f}

_{α}

*|(0; x) ∩ E* ^{0} *|*

^{0}

*f* *α* *(x)* = *α* 0; _{α} ^{x}

_{α}

^{x}

*∩* *α* ^{1} *E* ^{0}

^{0}

*f* ^{x} _{α} ^{x→0+} *−→ 0,*

^{x}

_{α}

^{x→0+}

*which implies 0 ∈ Φ* ^{+} *f*

*α*

*(E).* _{■}

### Corollary 7 *If Φ* *f* *∈* */* n

### Φ _{hsi} *; hsi ∈ e* *S* o

_{hsi}

*then Φ* *f*

*α*

*∈* */* n

### Φ _{hsi} *; hsi ∈ e* *S* o

_{hsi}

*for any α > 0.*

### Theorem 8 *If f is the function defined in Example 5 then Φ* *f*

*α*

### ⊈ Φ ^{f}

^{f}

*β*

*for any* *α > β > 1.*

*Proof Since f is nondecreasing, we have f* *α* *¬ f* ^{β} and Φ *f*

^{β}

*α*

*⊂ Φ* ^{f}

^{f}

*β*

*. Let (n* *i* ) be an *increasing sequence of positive integers such that r* *n*

*i*

*= 1 for every i. Define*

*E =* *ℝ \* [ *∞* *i=1*

*βb* *n*

*i*

*; βb* *n*

*i*

*√* *w* *n*

*i*

### .

### It is sufficient to show that

*0 ∈ Φ* ^{f}

^{f}

*β*

*(E) \ Φ* ^{f}

^{f}

*α*

*(E) .*

*Without loss of generality, we can assume that for any n* *βa* *n* *< b* *n* *and βb* *n* *< αb* *n* *< βb* *n* *√* *w* *n* *< a* *n* *−1* . *If x ∈ (βb* ^{n}

^{n}

*i*

*; βa* *n*

*i*

*−1* *] for some i then*

### (1) *|(0; x) ∩ E* ^{0} *|*

^{0}

*f* *β* *(x)* *¬* *βb* *n*

*i*

*√* *w* *n*

*i*

*f*

*x* *β*

### = *βb* *n*

*i*

*√* *w* *n*

*i*

*a* *n*

*i*

*−1*

### = *β*

*√* *w* *n*

*i*

### .

*If x ∈ (βa* ^{n}

^{n}

^{i}*; βb* *n*

*i*

*] for some i then*

### (2) *|(0; x) ∩ E* ^{0} *|*

^{0}

*f* *β* *(x)* *<* *a* *n*

*i*

*f*

*x* *β*

### = *a* *n*

*i*

*r* *n*

*i*

*b* *n*

*i*

### = 1 *w* *n*

*i*

### .

*If x ∈ (βa* ^{p} *; βb* *p* *−1* *] and n* *i* *< p < n* *i+1* *for some i then*

^{p}

### (3) *|(0; x) ∩ E* ^{0} *|*

^{0}

*f* *β* *(x)* *<* *a* *p*

*r* *p* *b* *p* *¬* 1 *w* *p*

### . *From (1)-(3) we conclude that 0 ∈ Φ* ^{f}

^{f}

*β*

*(E).*

*On the other hand for x* *i* *= αb* *n*

*i*

*|(0; x* ^{i} *) ∩ E* ^{0} *|*

^{i}

^{0}

*f* *α* *(x* *i* ) ** *|(βb* ^{n}

^{n}

^{i}*; αb* *n*

*i*

*)|*

*f (b* *n*

*i*

### ) = *(α − β) b* ^{n}

^{n}

^{i}*r* *n*

*i*

*b* *n*

*i*

*= α − β > 0,*

*and consequently 0 /* *∈ Φ* ^{f}

^{f}

^{α}*(E).* _{■}

*In [4, Th. 1] it has been proved that for each function f ∈ A there is a continuous* *function g ∈ A such that Φ* ^{f} = Φ *g* *. Thus there is at most continuum different f-* density topologies. Combining this result with Corollary 4, Corollary 7 and Theorem 8 we obtain

^{f}

### Corollary 9 *The family*

*T* ^{f} *; f ∈ A* ^{1}

^{f}

*\* n

*T* *hsi* *; hsi ∈ e* *S* o

*has cardinality continuum.*

### In the next theorem we will compare densities generated by sequences. Let us *formulate a useful lemma from [5]. Fix hai and hbi from e* *S. There is an unique* *sequence (k* *n* ) of positive integers such that

*b* *n* *∈ (a* ^{k}

^{k}

*n*

### +1 *; a* *k*

*n*

### ]

*for each n with b* *n* *< a* 1 *. From now on (k* *n* ) will denote this unique sequence.

### Lemma 10 *([5, Th. 7]). The following conditions are equivalent* *(a) Φ* _{hai} *⊂ Φ* *hbi* *.*

_{hai}

*(b) For arbitrary increasing sequence (n* *i* ) *of positive integers* lim inf

*i→∞*

*a* *k*

_{ni}*b* *n*

*i*

*<* *∞ or lim inf* _{i→∞} *b* *n*

_{i→∞}

*i*

*a* *k*

_{ni}### +1 = 1.

### Theorem 11 *Let hai , hbi ∈ e* *S. If Φ* *hbi* ⊈ Φ *hai* *then there is f ∈ A* ^{1} *such that* Φ _{hbi} *⊂ Φ* ^{f} *⊂ Φ* *hai* *and Φ* *f* *6= Φ* *hsi* *for hsi ∈ e* *S.*

_{hbi}

^{f}

*Proof By Proposition 1 we can asume that hai is a subsequence of hbi. Since* Φ _{hai} ⊈ Φ *hbi* *, there exists an increasing sequence (n* *i* ) of positive integers such that

_{hai}

*i* lim *→∞*

*a* *k*

_{ni}*b* *n*

*i*

*= ∞ and M = lim inf* _{i}

_{i}

*→∞*

*b* *n*

*i*

*a* *k*

_{ni}### +1 *> 1.*

### Let us denote

*β =*

### *√*

4
*M* *; M < ∞*

### 2 *; M = ∞* .

*Replacing (n* _{i} *) by a subsequence we can assume that (k* _{n}

_{i}

_{n}

_{i}### ) is increasing and for *every i*

### (4) ^{a} _{b}

^{a}

_{b}

^{kni}*ni*

*> i* ^{2} , _{a} ^{b}

_{a}

^{b}

^{ni}*kni +1*

*> β* ^{3} and _{a} ^{b}

_{a}

^{b}

^{ni}*kni +1*

*< β* ^{5} *if M < ∞.*

### Let us define

*hci = hai ∪ (b* ^{n}

^{n}

*i*

### ) ,

*hri = (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) ,* *g* 1 *(x) = a* *n* *for x ∈ (a* ^{n+1} *; a* *n* ] , *g* 2 *(x) = c* *n* *for x ∈ (c* ^{n+1} *; c* *n* ] , *g* 3 *(x) = b* *n* *for x ∈ (b* ^{n+1} *; b* *n* ] ,

^{n+1}

^{n+1}

^{n+1}

*f (x) =*

### *r* *i* *b* *n*

*i*

*for x ∈ a* ^{k}

^{k}

_{ni}^{+1} *; b* *n*

*i*

### , *g* 1 *(x) for x /* *∈* S *∞*

*i=1* *a* *k*

_{ni}### +1 *; b* *n*

*i*

### .

*Since 1 ¬ r* *i* *¬ i* ^{2} *, f ∈ A* ^{1} *and g* _{3} *¬ g* ^{2} *¬ f ¬ g* ^{1} . Thus

### Φ _{hbi} = Φ *g*

_{hbi}

_{3}

*⊂ Φ* *hci* = Φ *g*

_{2}

*⊂ Φ* ^{f} *⊂ Φ* ^{g}

1 ^{f}

^{g}

### = Φ _{hai} .

_{hai}

### We will show that Φ *f* *6= Φ* *hsi* *for hsi ∈ e* *S. Suppose, contrary to our claim, that there* *is hsi ∈ e* *S such that*

### (5) Φ *f* = Φ _{hsi} .

_{hsi}

### Let us consider the sets

*T* =

*i* *∈ ℕ; ∃* ^{m} *∈ℕ* *s* *m* *∈*

^{m}

*β* ^{3} *a* *k*

_{ni}### +1 *; ib* *n*

*i*

### , *R* *= {r* ^{i} *; i ∈ T } .*

^{i}

*We will prove that the set R is bounded. Suppose it is not true and so T is infinite.*

*Let (t* *i* *) be an increasing sequence consisting of all elements from T , i.e.*

*T =* *{t* ^{i} *; i ∈ ℕ} .*

^{i}

*Let m* *i* be a fixed positive integer such that *s* *m*

*i*

*∈* h

*β* ^{3} *a* *k*

_{nti}### +1 *; t* *i* *b* *n*

_{ti}### i .

### We will show that

### (6) *s* *m*

*i*

*< 2b* *n*

_{ti}*for almost every i. Suppose on the contrary that there is an increasing sequence (p* *i* ) *of positive integers such s* *m*

_{pi}* 2b* ^{n}

^{n}

_{tpi}*for every i. Hence*

*a* *k*

_{ntp}*i*

*s* *m*

_{pi}** *t* ^{2} _{p}

_{p}

_{i}*b* *n*

_{tpi}*t* *p*

*i*

*b* *n*

_{tpi}*= t* *p*

*i*

* p* ^{i} * i*

^{i}

### and so

*i* lim *→∞*

*a* *k*

_{ntp}*i*

*s*

_{mpi}*= ∞* and

### lim inf

*i* *→∞*

*s*

_{mpi}*b* *n*

_{tpi}* 2.*

### This, by Lemma 10, contradicts the assumption that Φ _{hci} *⊂ Φ* *hsi* and finishes the proof of (6).

_{hci}

*Now we show that there are real numbers ε > δ > 0 such that* (7) *a* *k*

_{nti}### +1 *< δb* *n*

_{ti}*< εb* *n*

_{ti}*< s* *m*

*i*

*for almost every i. If M = lim inf*

*i* *→∞*

*b*

_{ni}*a*

_{kni +1}*<* *∞ then by (4) it is sufficient to set* *ε =* _{β} ^{1}

2 _{β}

*and δ =* _{β} ^{1}

3_{β}

*. If M = ∞, then obviously it is sufficient to show that for some* *positive ε inequality*

*εb* *n*

_{ti}*< s* *m*

*i*

*holds for almost every i. Suppose on the contrary that there is an increasing sequence* *(p* *i* *) of positive integers such is* *m*

_{pi}*< b* *n*

_{tpi}*for every i. Hence*

*i* lim *→∞*

*b* *n*

_{tpi}*s* *m*

_{pi}*= ∞.*

### and

### lim inf

*i* *→∞*

*s* *m*

_{pi}*a* *k*

_{ntp}*i*

### +1 * β* ^{3} *> 1.*

### which also contradicts the assumption that Φ _{hci} *⊂ Φ* *hsi* and ends the proof of (7).

_{hci}

*By our assumptions, the set R is not bounded. Thus there exists an increasing* *sequence (p* *i* ) of positive integers such that

*i→∞* lim *r* *t*

_{pi}*= ∞.*

### Put

*A =* [ *∞* *i=1*

### *δb* *n*

_{tpi}*; εb* *n*

_{tpi}### .

### From (6) and (7) it follows that *A ∩ 0; s* *m*

_{pi}*s* *m*

_{pi}** *εb* *n*

_{tpi}*− δb* ^{n}

^{n}

_{tpi}*2b* *n*

_{tpi}### = *ε* *− δ* 2 *> 0* *for almost every i, and so*

### (8) *0 /* *∈ Φ* ^{+} _{hsi} *(A* ^{0} ).

_{hsi}

^{0}

*We will show that 0 ∈ Φ* ^{+} *f* *(A* ^{0} *). Let x ∈ (a* ^{j} *; a* *j* *−1* *]. If j = k* *n*

^{0}

^{j}

_{tpi}*for some i then*

### (9) *|A ∩ (0; x)|*

*f (x)* *¬* *εb* *n*

_{tpi}*r* *t*

_{pi}*b* *n*

_{tpi}### = *ε* *r* *t*

_{pi}### .

*On the other hand if k* *m*

_{ni}*< j < k* *m*

_{ni+1}*for some i then*

### (10) *|A ∩ (0; x)|*

*f (x)* *¬* *εb* *n*

_{tpi+1}*a* *j+1* *¬* *εb* *n*

_{tpi+1}*a* *k*

_{ntp}*i+1*

*¬* *ε*

*t* ^{2} _{p}

_{p}

_{i+1}*¬* *ε* *(i + 1)* ^{2} .

*From (5), (9) and (10) it follows that 0 ∈ Φ* ^{+} *f* *(A* ^{0} ) = Φ ^{+} _{hsi} *(A* ^{0} ), contrary to (8). Hence *we conclude that R is bounded.*

^{0}

_{hsi}

^{0}

*Let L be a positive integer such that* *L > sup R,*

*(p* *i* ) be an increasing sequence of positive integers for which *r* *p*

*i*

*= L*

### and

*B =* [ *∞* *i=1*

### 1

*β* *b* *n*

_{pi}*; b* *n*

_{pi}### .

### Since

*B ∩ 0; b* ^{n}

^{n}

_{pi}*f (b* *n*

_{pi}### ) ** *b* *n*

_{pi}*−* *β* ^{1} *b* *n*

_{pi}*r* *p*

*i*

*b* *n*

_{pi}### = *1 −* ^{1} *β*

*L* *> 0,*

### (11) *0 /* *∈ Φ* ^{+} *f* *(B* ^{0} ).

^{0}

*We will prove that 0 ∈ Φ* ^{+} _{hsi} *(B* ^{0} *). By definition of (p* *i* *), p* *i* *∈ T for every i and* */* consequently

_{hsi}

^{0}

*s* *m* *∈* */* h

*β* ^{3} *a* *k*

_{npi}### +1 *; p* *i* *b* *n*

_{pi}### i

*for m, i ∈ ℕ. Let us consider any term s* ^{m} *of hsi. There is a positive integer j such* *that s* *m* *∈ (a* ^{j} *; a* _{j−1} *]. If j = k* *n*

^{m}

^{j}

_{j−1}

_{pi}*for some i then*

*s* *m* *< β* ^{3} *a* *k*

_{npi}### +1 *or s* *m* *> p* *i* *b* *n*

_{pi}### . In the first case we have

### (12) *|B ∩ (0; s* ^{m} *)|*

^{m}

*s* *m* *¬* *b* *n*

_{pi+1}*a* *k*

_{npi}### +1 *¬* *b* *n*

_{pi+1}*a* *k*

_{npi+1}*¬* 1

*p* ^{2} _{i+1} *¬* 1 *(i + 1)* ^{2} , and in the second

_{i+1}

### (13) *|B ∩ (0; s* ^{m} *)|*

^{m}

*s* *m* *¬* *b* *n*

_{pi}*p* *i* *b* *n*

_{pi}*¬* 1 *i* . *Moreover, if k* *n*

_{pi}*< j < k* *n*

_{pi+1}### , then

### (14) *|B ∩ (0; s* ^{m} *)|*

^{m}

*s* *m* *¬* *b* *n*

_{pi+1}*a* *j+1* *¬* *b* *n*

_{pi+1}*a* *k*

_{npi+1}*¬* 1

*p* ^{2} _{i+1} *¬* 1 *(i + 1)* ^{2} .

_{i+1}

*From (5) and (12)-(14) it follows that 0 ∈ Φ* ^{+} _{hsi} *(B* ^{0} ) = Φ ^{+} _{f} *(B* ^{0} ), contrary to (11). _{■}

_{hsi}

^{0}

_{f}

^{0}

### Corollary 12 *If T* *hbi* *⊈ T* *hai* *then there is f ∈ A* ^{1} *such that T* *hbi* *⊂ T* ^{f} *⊂ T* *hai* *and* *the topology T* ^{f} *is generated by no sequence.*

^{f}

^{f}